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Nordic 2025 P3
anirbanbz 8
N
by lksb
Source: Nordic 2025
Let 
be an acute triangle with orthocenter 
and circumcenter 
. Let 
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
be an acute triangle with orthocenter 
and circumcenter 
. Let 
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
8 replies
another functional inequality?
Scilyse 32
N
by ihategeo_1969
Source: 2023 ISL A4
Let 
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](//latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
32 replies
Mount Inequality erupts in all directions!
BR1F1SZ 1
N
by sami1618
Source: Austria National MO Part 1 Problem 1
Let 
, 
and 
be pairwise distinct nonnegative real numbers. Prove that
![\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]](//latex.artofproblemsolving.com/2/0/9/209f9807481e7a1e97f023bf65a603cb462c62cb.png)
(Karl Czakler)
, 
and 
be pairwise distinct nonnegative real numbers. Prove that![\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]](http://latex.artofproblemsolving.com/2/0/9/209f9807481e7a1e97f023bf65a603cb462c62cb.png)
(Karl Czakler)
1 reply
Division involving difference of squares
BR1F1SZ 1
N
by grupyorum
Source: Austria National MO Part 1 Problem 4
Determine all integers 
that can be written in the form
![\[
n = \frac{a^2 - b^2}{b},
\]](//latex.artofproblemsolving.com/f/9/0/f90a3ce2e1b51d6e1cb53a15cad6f36c2b30492a.png)
where and are positive integers.
(Walther Janous)
that can be written in the form![\[
n = \frac{a^2 - b^2}{b},
\]](http://latex.artofproblemsolving.com/f/9/0/f90a3ce2e1b51d6e1cb53a15cad6f36c2b30492a.png)
where and are positive integers.(Walther Janous)
1 reply
Erasing the difference of two numbers
BR1F1SZ 0
Source: Austria National MO Part 1 Problem 3
Consider the following game for a positive integer . Initially, the numbers 
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
, or 
as 
, or 
as 
.)
For which values of is it possible to end with only one number remaining on the board?
(Michael Reitmeir)
. Initially, the numbers 
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
, or 
as 
, or 
as 
.)For which values of
is it possible to end with only one number remaining on the board?(Michael Reitmeir)
0 replies
4 wise men and 100 hats. 3 must guess their numbers
NO_SQUARES 2
N
by NO_SQUARES
Source: 239 MO 2025 10-11 p5
There are four wise men in a row, each sees only those following him in the row, i.e. the 
st sees the other three, the 
nd sees the rd and 
th, and the rd sees only the th. The devil has 
hats, numbered from to , he puts one hat on each wise man, and hides the extra 
hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
st sees the other three, the 
nd sees the rd and 
th, and the rd sees only the th. The devil has 
hats, numbered from to , he puts one hat on each wise man, and hides the extra 
hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
2 replies
\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}
NO_SQUARES 2
N
by ektorasmiliotis
Source: 239 MO 2025 8-9 p4
Positive numbers , and are such that 
. Prove that ![\[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]](//latex.artofproblemsolving.com/3/e/4/3e4b336137b17e55bf172d45bd18fe6d5561a827.png)
, and are such that 
. Prove that ![\[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]](http://latex.artofproblemsolving.com/3/e/4/3e4b336137b17e55bf172d45bd18fe6d5561a827.png)
2 replies
BMO 2024 SL A4
MuradSafarli 2
N
by GreekIdiot
A4.
Let
be real numbers such that 
.
Prove that:
![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](//latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
Let
be real numbers such that 
.Prove that:
![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
2 replies
Aime type Geo
ehuseyinyigit 0
Source: Turkish First Round 2024
In a scalene triangle , let 
be the midpoint of side 
. Let the line perpendicular to at point 
intersect 
at 
. If 
is tangent to at 
, find 
.
, let 
be the midpoint of side 
. Let the line perpendicular to at point 
intersect 
at 
. If 
is tangent to at 
, find 
.
0 replies
1996 St. Petersburg City Mathematical Olympiad
Sadece_Threv 2
N
by reni_wee
Source: 1996 St. Petersburg City Mathematical Olympiad
Find all positive integers such that 
divides 
such that 
divides 
2 replies
